Question: Find the minimum value of
\[\sqrt{x^2 + (1 - x)^2} + \sqrt{(1 - x)^2 + (1 + x)^2}\]over all real numbers $x.$
Answer: In the coordinate plane, let $A = (0,1),$ $B = (1,-1),$ and $P = (x,x).$  Then
\[AP = \sqrt{x^2 + (1 - x)^2}\]and
\[BP = \sqrt{(x - 1)^2 + (x + 1)^2},\]so we want to minimize $AP + BP,$ subject to $P$ lying on the line $y = x.$

[asy]
unitsize(2.5 cm);

pair A, B, P;

A = (0,1);
B = (1,-1);
P = (0.8,0.8);

draw(A--P--B);
draw((-0.2,-0.2)--(1.2,1.2),dashed);

label("$y = x$", (1.2,1.2), NE);

dot("$A$", A, NW);
dot("$B$", B, SE);
dot("$P$", P, N);
[/asy]

By the Triangle Inequality, $AP + BP \ge AB = \sqrt{5}.$  Equality occurs when $P$ is the intersection of the line $y = x$ and line $AB$ (which occurs when $x = \frac{1}{3}$), so the minimum value is $\boxed{\sqrt{5}}.$